A-V Formulation using "regularized" Method with Gauge Condition

In the section General Case, Maxwell equation’s formulation is computed in a Maxwell Quasi Static approximation, the A-V Formulation with an imposed gauge condition on potential magnetic field :

\[ \nabla \cdot \mathbf{A} = 0\]

1. Transient Case

1.1. A-V Formulation

This section recalls the A-V Formulation :

Thus \(\Omega\) the domain, comprising the conductor (or superconductor) domain \(\Omega_c\) and non conducting materials \(\Omega_n\) (\(\mathbf{J} = 0\)) like the air for example. \(\Gamma = \partial \Omega\) is the boundary of \(\Omega\), \(\Gamma_c = \partial \Omega_c\) the boundary of \(\Omega_c\), \(\Gamma_D\) the boundary with Dirichlet boundary condition and \(\Gamma_N\) the boundary with Neumann boundary condition, such that \(\Gamma = \Gamma_D \cup \Gamma_N\).

We introduce :

  • Magnetic potential field \(\mathbf{A}\) : \(\textbf{B} = \nabla \times \textbf{A}\)

  • Electric potential scalar : \(\nabla V = - \textbf{E} - \frac{\partial \textbf{A}}{\partial t}\)

We want to resolve the electromagnetism problem ( with \(\mathbf{A}\) and \(V\) the unknows) :

A-V Formulation
\[\text{(AV)} \left\{ \begin{matrix} \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \sigma \frac{\partial \textbf{A}}{\partial t} + \sigma \nabla V &=& 0 \text{ on } \Omega & \text{(AV-1)} \\ \nabla \cdot \left( \sigma \nabla V + \sigma \frac{\partial \textbf{A}}{\partial t} \right) &=& 0 \text{ on } \Omega_c & \text{(AV-2)} \\ \mathbf{A} \times \mathbf{n} &=& 0 \text{ on } \Gamma_D & \text{(DA)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 \text{ on } \Gamma_N & \text{(NA)} \\ V &=& V_0 \text{ on } \Gamma_{DV} & \text{(DV)} \\ \frac{\partial V}{\partial \mathbf{n}} &=& 0 \text{ on } \Gamma_{NV} & \text{(NV)} \end{matrix} \right.\]

1.2. Differential Formulation

In order to keep the curlcurl formulation and guarantee the unicity of the solution of our problem, we will use the Regularized formulation as shown in Cecile Daversin - Catty. Reduced basis method applied to large non-linear multi-physics problems:application to high field magnets design. Electromagnetism. Université de Strasbourg, 2016. :

A-V Formulation
\[ \text{(AV)} \left\{ \begin{matrix} \sigma \frac{\partial \textbf{A}}{\partial t} + \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \epsilon \textbf{A} + \sigma \nabla V &=& 0 &\text{ on } \Omega & \text{(AV-1)} \\ \nabla \cdot \left( \sigma \nabla V + \sigma \frac{\partial \textbf{A}}{\partial t} \right) &=& 0 &\text{ on } \Omega_c & \text{(AV-2)} \\ \mathbf{A} \times \mathbf{n} &=& 0 &\text{ on } \Gamma_D & \text{(DA)} \\ \left( \nabla \times \mathbf{A} \right) \times \mathbf{n} &=& 0 &\text{ on } \Gamma_N & \text{(NA)} \\ V &=& V_0 &\text{ on } \Gamma_{DV} & \text{(DV)} \\ \frac{\partial V}{\partial \mathbf{n}} &=& 0 &\text{ on } \Gamma_{NV} & \text{(NV)} \end{matrix} \right.\]

With \(\epsilon\) the weight of the regularization term.

1.3. Weak Formulation

The weak formulation of equation (AV Regularized Formulation) can be expressed as follows :

We introduce the set :

\[H^{curl}(\Omega) = \{ \textbf{v} \in L^2(\Omega), \nabla \times \textbf{v} \in L^2(\Omega) \}\]

The set \(H^{curl}(\Omega)\) is a Hilbert space with the scalar product :

\[<\mathbf{v_1},\mathbf{v_2}>_{H^{curl}(\Omega)} = \int_{\Omega}{ \mathbf{v_1} \cdot \mathbf{v_2} + \nabla \times \mathbf{v_1} \cdot \nabla \times \mathbf{v_2} } \text{ for all } \mathbf{v_1}, \mathbf{v_2} \in H^{curl}(\Omega)\]

We multiply the equation by \(\phi \in H^{curl}(\Omega)\) and integrate it on \(\Omega\) :

\[ \int_{\Omega}{ \left(\sigma \frac{\partial \textbf{A}}{\partial t} + \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \epsilon \textbf{A} \right) \cdot \phi \ dxdydz} + \int_{\Omega_c}{ \sigma \nabla V \cdot \phi \ dxdydz} = 0 \hspace{1cm} \text{(Weak AV)}\]

By Formula of Green :

\[\scriptsize{ \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A}\cdot \nabla \times \phi \ dxdydz } - \int_{\Gamma}{ \frac{1}{\mu} \frac{\partial \mathbf{A}}{\partial \mathbf{n}} \cdot \phi }+ \int_{\Omega}{ \epsilon\textbf{A} \cdot \phi \ dxdydz }+ \int_{\Omega_c}{ \sigma \nabla V \cdot\phi \ dxdydz } = 0 \hspace{1cm} \text{(Disc AV)} }\]

The imposed boundary conditions are :

  • Dirichlet : \(\mathbf{A} \times \mathbf{n} = 0\) on \(\Gamma_D\)

  • Neumann : \(\left( \nabla \times \mathbf{A} \right) \times \mathbf{n} = 0\) on \(\Gamma_N\)

So, the weak formulation is :

Weak formulation of A-V Regularized Formulation
\[\scriptsize{ \text{(Weak AV)} \\ \left\{ \begin{eqnarray*} \int_{\Omega}{ \sigma \frac{\partial \textbf{A}}{\partial t} \cdot \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A} \cdot \nabla \times \phi \ dxdydz } + \int_{\Omega}{ \epsilon \textbf{A} \cdot \phi \ dxdydz } &=& - \int_{\Omega_c}{ \sigma \nabla V \cdot\phi \ dxdydz } \quad\text{(Weak AV1)} \\ \int_{\Omega_c}{\sigma (\nabla V + \frac{\partial \mathbf{A}}{\partial t}) \cdot \nabla \mathbf{\psi}} &=& 0 \quad\text{(Weak AV-2)}\\ &&\forall \mathbf{\phi} \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{eqnarray*} \right. }\]

1.4. Time Discretization

In this subsection, we use the time discretization by backward Euler method on the (Weak AV 1).

We discretize in time the problem with the time step \(\Delta t\).

We note \(f^n(\mathbf{x}) = f(n\Delta t, \mathbf{x})\), for \(n \in \mathbb{N}\).

We have the approximation with backward Euler method : \(\frac{\partial A}{\partial t} \approx \frac{A^{n+1}-A^n}{\Delta t}\).

The equation (Weak AV 1) becomes :

Time Discretization of A-V Regularized Formulation
\[\scriptsize{ \begin{eqnarray*} \int_{\Omega}{ \sigma \frac{\textbf{A}^{n+1}}{\Delta t} \cdot \phi \ dxdydz} + \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A}^{n+1} \cdot \nabla \times \phi \ dxdydz } + \int_{\Omega}{ \epsilon \textbf{A}^{n+1} \cdot \phi \ dxdydz } &=& - \int_{\Omega_c}{ \sigma \nabla V^{n+1} \cdot\phi \ dxdydz } +\int_{\Omega}{ \sigma \frac{\textbf{A}^{n}}{\Delta t} \cdot \phi \ dxdydz} \end{eqnarray*}}\]

2. Magnetostatic Case

In a stationary case, \(\frac{\partial f}{\partial t} = 0\).

2.1. Differential Equation

The differential formulation of A-V Formulation becomes :

A-V Regularized Formulation in Stationary Case
\[\text{(Magstat Regularized)} \left\{ \begin{matrix} \nabla \times \left( \frac{1}{\mu} \nabla \times \textbf{A} \right) + \epsilon \textbf{A} + \sigma \nabla V &=& 0 &\text{ on } \Omega & \text{(AV-1)} \\ \nabla \cdot \left( \sigma \nabla V \right) &=& 0 &\text{ on } \Omega_c & \text{(AV-2)} \\ \end{matrix} \right.\]

2.2. Weak Formulation

The weak formulation of A-V Formulation becomes :

Weak formulation of A-V Regularized Formulation in Stationary
\[\scriptsize{ \text{(Weak Magstat Regularized)} \\ \left\{ \begin{eqnarray*} \int_{\Omega}{ \frac{1}{\mu} \nabla \times \textbf{A} \cdot \nabla \times \phi \ dxdydz } + \int_{\Omega}{ \epsilon \textbf{A} \cdot \phi \ dxdydz } &=& - \int_{\Omega_c}{ \sigma \nabla V \cdot\phi \ dxdydz } \quad\text{(Weak AV1)} \\ \int_{\Omega_c}{\sigma \nabla V \cdot \nabla \mathbf{\psi}} &=& 0 \quad\text{(Weak AV-2)}\\ &&\forall \phi \in H^{curl}(\Omega) \text{ and } \forall \mathbf{\psi} \in H^{1}(\Omega_c) \end{eqnarray*} \right. }\]

3. References

  • Cecile Daversin - Catty. Reduced basis method applied to large non-linear multi-physics problems: application to high field magnets design. Electromagnetism. Université de Strasbourg, 2016. p56-64 PDF